Math Problem Statement

Matritsaning chiziqli kombinatsiyasini toping

Matritsaning chiziqli kombinatsiyasini toping

Solution

To solve the problem, we need to compute A+BTA + B^T, where AA and BB are given matrices. Let’s go through the steps:

Step 1: Define the Matrices

Matrix AA and BB are defined as: A=(710172027),B=(312151214)A = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}

Step 2: Find the Transpose of BB (BTB^T)

The transpose of BB is obtained by swapping rows and columns: BT=(312151214)T=(312151214)B^T = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}^T = \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}

Step 3: Add AA and BTB^T

Now, we perform element-wise addition of AA and BTB^T: A+BT=(710172027)+(312151214)A + B^T = \begin{pmatrix} 7 & -1 & 0 \\ -1 & 7 & -2 \\ 0 & -2 & 7 \end{pmatrix} + \begin{pmatrix} 3 & 1 & 2 \\ 1 & -5 & 1 \\ 2 & 1 & -4 \end{pmatrix}

Step 4: Compute the Result

Adding the matrices element-wise: A+BT=(7+31+10+21+1752+10+22+174)A + B^T = \begin{pmatrix} 7 + 3 & -1 + 1 & 0 + 2 \\ -1 + 1 & 7 - 5 & -2 + 1 \\ 0 + 2 & -2 + 1 & 7 - 4 \end{pmatrix} =(1002021213)= \begin{pmatrix} 10 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}

Final Answer

A+BT=(1002021213)A + B^T = \begin{pmatrix} 10 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & -1 & 3 \end{pmatrix}

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

  1. How do you find the transpose of a matrix, and what properties does it have?
  2. What is an element-wise addition of matrices, and when is it defined?
  3. How does matrix addition relate to matrix multiplication?
  4. Can we generalize this operation for matrices of different sizes?
  5. How would ABTA - B^T differ in computation and result from A+BTA + B^T?

Tip: Always check matrix dimensions to ensure they match when performing operations like addition and multiplication.

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Math Problem Analysis

Mathematical Concepts

Matrix addition
Matrix transpose
Linear combinations

Formulas

Matrix addition: C = A + B where C_ij = A_ij + B_ij
Matrix transpose: B^T where (B^T)_ij = B_ji

Theorems

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Suitable Grade Level

College level